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2016 Fermat Contest
Solutions
(Grade 11)

Wednesday, February 24, 2016
(in North America and South America)

Thursday, February 25, 2016
(outside of North American and South America)

©2015 University of Waterloo


  1. Since x=3 and y=2x, then y=23=6.
    Since y=6 and z=3y, then z=36=18.

    Answer: (D)

  2. A square-based pyramid has 8 edges: 4 edges that form the square base and 1 edge that joins each of the four vertices of the square base to the top vertex.

    Answer: (C)

  3. Evaluating, 20+16×2020×16=20+320320=340320=1716.
    Alternatively, we could notice that each of the numerator and denominator is a multiple of 20, and so 20+16×2020×16=20(1+16)20×16=1+1616=1716.

    Answer: (E)

  4. The 7th oblong number is the number of dots in retangular grid of dots with 7 columns and 8 rows.
    Thus, the 7th oblong number is 7×8=56.

    Answer: (C)

  5. Solution 1
    Let the coordinates of R be (a,b).
    Since Q is the midpoint of P and R, then the difference between the x-coordinates of Q and P equals the difference between the x-coordinates of R and Q.
    In other words, a4=41 and so a=7.
    Similarly, b7=73 and so b=11.
    Thus, the coordinates of R are (7,11).

    Solution 2
    Let the coordinates of R be (a,b).
    The midpoint of P(1,3) and R(a,b) has coordinates (12(1+a),12(3+b)).
    Since Q(4,7) is the midpoint of PR, then 4=12(1+a) (which gives 7=1+a or a=7) and 14=12(3+b) (which gives 14=3+b or b=11).
    Therefore, the coordinates of R are (7,11).

    Answer: (B)

  6. Each week, Carrie sends 5 messages to her brother on each of 2 days, for a total of 10 messages.
    Each week, Carrie sends 2 messages to her brother on each of the remaining 5 days, for a total of 10 messages.
    Therefore, Carrie sends 10+10=20 messages per week.
    In four weeks, Carrie sends 420=80 messages.

    Answer: (D)

  7. Evaluating, (2)3(3)2=89=17.

    Answer: (A)

  8. Since 25n=3, then 25n=9.
    Thus, n=16 and so n=162=256.

    Answer: (E)

  9. Solution 1
    Since x% of 60 is 12, then x10060=12 or x=1210060=20.
    Therefore, 15% of x is 15% of 20, or 0.1520=3.

    Solution 2
    Since x% of 60 is 12, then x10060=12 or 60x100=12.
    In terms of x, 15% of x equals 15100x or 15x100.
    Since 60x100=12, then 15x100=14(60x100)=1412=3.

    Answer: (D)

  10. Solution 1
    Since square PQRS has side length 2, then PQ=QR=RS=SP=2.
    Since W, X, Y, Z are the midpoints of the sides of PQRS, then PW=PZ=1.
    Since ZPW=90, then WZ=PW2+PZ2=12+12=2.
    Therefore, square WXYZ has side length 2.
    The area of square WXYZ is (2)2=2 and the area of square PQRS is 22=4.
    The ratio of these areas is 2:4 or 1:2.

    Solution 2
    Join W to Y and X to Z.

    Since PQRS is a square and W, X, Y, and Z are the midpoints of its sides, then WY and ZX divide the square into four identical squares.
    Each of these four squares is divided into two triangles of equal area by its diagonal. (These diagonals are WZ, WX, XY, YZ.)
    Square WXYZ is made up of 4 of these triangles of equal area.
    Square PQRS is made up of 8 of these triangles of equal area.
    Therefore, the ratio of these areas is 4:8 or 1:2.

    Answer: (A)

  11. By the Pythagorean Theorem in PRS, PS=RS2PR2=132122=169144=25=5 since PS>0.
    Thus, PQ=PS+SQ=5+11=16.
    By the Pythagorean Theorem in PRQ, RQ=PR2+PQ2=122+162=144+256=400=20 since RQ>0.
    Therefore, the perimeter of QRS is RS+SQ+RQ=13+11+20=44.

    Answer: (B)

  12. Since 128=27, its positive divisors are 20=121=222=423=824=1625=3226=6427=128 Of these, the integers 1,4,16,64 are perfect squares, which means that 128 has three positive divisors that are perfect squares larger than 1.

    Answer: (D)

  13. Since 4x,2x3,4x3 form an arithmetic sequence, then the differences between consecutive terms are equal, or (2x3)4x=(4x3)(2x3).
    Thus, 2x3=2x or 4x=3 and so x=34.

    Answer: (E)

  14. Since the average of the four numbers 4,a,b,22 is 13, then 4+a+b+224=13 and so 4+a+b+22=52 or a+b=26.
    Since a>4 and a is an integer, then a5.
    Since a+b=26 and a<b, then a is less than half of 26, or a<13.
    Since a is an integer, then a12.
    Therefore, we have 5a12.
    There are 8 choices for a in this range: 5, 6, 7, 8, 9, 10, 11, 12. (Note that 125+1=8.)
    These give the pairs (a,b)=(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14).
    Thus, there are 8 possible pairs.

    Answer: (B)

  15. When Hicham runs 10 km at an average speed of 12 km/h, he takes 1012=56 hours to run this distance.
    Since Hicham runs for a total of 1.5 hours, then he runs the last 6 km in 3256=9656=46=23 hours.
    Since he runs 6 km in 23 hours, his average speed for this segment is 62/3=9 km/h.

    Answer: (B)

  16. Solution 1
    Since x=18 is a solution to the equation x2+12x+c=0, then x=18 satisfies this equation.
    Thus, 182+12(18)+c=0 and so 324+216+c=0 or c=540.
    Therefore, the original equation becomes x2+12x540=0 or (x18)(x+30)=0.
    Therefore, the other solution is x=30.

    Solution 2
    We use the fact that the sum of the roots of an equation of the form x2+bx+c=0 is b.
    If the roots of the equation x2+12x+c=0 are 18 and r, then 18+r=12 or r=30.
    Therefore, the other solution is x=30.

    Answer: (C)

  17. The number of points on the circle equals the number of spaces between the points around the circle.
    Moving from the point labelled 7 to the point labelled 35 requires moving 357=28 points and so 28 spaces around the circle.
    Since the points labelled 7 and 35 are diametrically opposite, then moving along the circle from 7 to 35 results in travelling halfway around the circle.
    Since 28 spaces makes half of the circle, then 228=56 spaces make the whole circle.
    Thus, there are 56 points on the circle, and so n=56.

    Answer: (C)

  18. The first equation xyx+y=9 gives xy=9x+9y and so 8x=10y or 4x=5y.
    The second equation xyx+y=60 gives xy=60x60y.
    Multiplying this equation by 5 gives 5xy=300x300y or x(5y)=300x60(5y).
    Since 5y=4x, then x(4x)=300x60(4x) or 4x2=60x.
    Rearranging, we obtain 4x260x=0 or 4x(x15)=0.
    Therefore, x=0 or x=15.
    Since y=45x, then y=0 or y=12.
    From the first equation, it cannot be the case that x=y=0.
    We can check that the pair (x,y)=(15,12) satisfies both equations.
    Therefore, (x+y)+(xy)+xy=3+27+(180)=150.

    Answer: (B)

  19. Solution 1
    Suppose that, when the n students are put in groups of 2, there are g complete groups and 1 incomplete group.
    Since the students are being put in groups of 2, an incomplete group must have exactly 1 student in it.
    Therefore, n=2g+1.
    Since the number of complete groups of 2 is 5 more than the number of complete groups of 3, then there were g5 complete groups of 3.
    Since there was still an incomplete group, this incomplete group must have had exactly 1 or 2 students in it.
    Therefore, n=3(g5)+1 or n=3(g5)+2.
    If n=2g+1 and n=3(g5)+1, then 2g+1=3(g5)+1 or 2g+1=3g14 and so g=15.
    In this case, n=2g+1=31 and there were 15 complete groups of 2 and 10 complete groups of 3.
    If n=2g+1 and n=3(g5)+2, then 2g+1=3(g5)+2 or 2g+1=3g13 and so g=14.
    In this case, n=2g+1=29 and there were 14 complete groups of 2 and 9 complete groups of 3.
    If n=31, dividing the students into groups of 4 would give 7 complete groups of 4 and 1 incomplete group.
    If n=29, dividing the students into groups of 4 would give 7 complete groups of 4 and 1 incomplete group.
    Since the difference between the number of complete groups of 3 and the number of complete groups of 4 is given to be 3, then it must be the case that n=31.
    In this case, n2n=31231=930; the sum of the digits of n2n is 12.

    Solution 2
    Since the n students cannot be divided exactly into groups of 2, 3 or 4, then n is not a multiple of 2, 3 or 4.
    The first few integers larger than 1 that are not divisible by 2, 3 or 4 are 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.
    In each case, we determine the number of complete groups of each size:

    n 5 7 11 13 17 19 23 25 29 31 35
    # of complete groups of 2 2 3 5 6 8 9 11 12 14 15 17
    # of complete groups of 3 1 2 3 4 5 6 7 8 9 10 11
    # of complete groups of 4 1 1 2 3 4 4 5 6 7 7 8

    Since the number of complete groups of 2 is 5 more than the number of complete groups of 3 which is 3 more than the number of complete groups of 4, then of these possibilities, n=31 works.
    In this case, n2n=31231=930; the sum of the digits of n2n is 12.
    (Since the problem is a multiple choice problem and we have found a value of n that satisfies the given conditions and for which an answer is present, then this answer must be correct. Solution 1 shows why n=31 is the only value of n that satisfies the given conditions.)

    Answer: (B)

  20. Since points Y, W and Q form a straight line segment, then YWV=180VWQ and so YWV=180125=55.
    Since Q is the final position of Q after folding, then QWV is the final position of QWV after folding, and so QWV=QWV.
    Thus, QWV=QWV=125 and so QWY=QWVYWV=12555=70.

    Since QW and RY are parallel sides of the piece of paper, then RYW+QWY=180, and so RYW=180QWY=18070=110.
    Finally, PYV is opposite RYW so PYV=RYW=110.

    Answer: (A)

  21. When a marble is chosen from Box 1, the probability is 12 that it will be gold and the probability is 12 that it will be black.
    Thus, after this choice is made, there is a probability of 12 that Box 2 contains 2 gold marbles and 2 black marbles, and there is a probability of 12 that Box 2 contains 1 gold marble and 3 black marbles.
    In the first case (which occurs with probability 12), the probability that a gold marble is chosen from Box 2 is 24=12 and the probability that a black marble is chosen from Box 2 is 24=12.
    In the second case (which occurs with probability 12), the probability that a gold marble is chosen from Box 2 is 14 and the probability that a black marble is chosen from Box 2 is 34.

    Therefore, the probability that a gold marble is chosen from Box 2 is 1212+1214=38 and the probability that a black marble is chosen from Box 2 is 1212+1234=58.
    Thus, after this choice is made, there is a probability of 38 that Box 3 contains 2 gold marbles and 3 black marbles, and a probability of 58 that Box 3 contains 1 gold marble and 4 black marbles.
    Finally, the probability that a gold marble is chosen from Box 3 equals the probability that Box 3 contains 2 gold marbles and 3 black marbles times the probability of choosing a gold marble in this situation (that is, 3825) plus the probability that Box 3 contains 1 gold marble and 4 black marbles times the probability of choosing a gold marble in this situation.
    In other words, this probability is 3825+5815=1140.

    Answer: (A)

  22. Solution 1
    We expand the given expression to obtain (x2+6x+9)+2(y24y+4)+4(x214x+49)+(y2+8y+16) We expand further to obtain x2+6x+9+2y28y+8+4x256x+196+y2+8y+16 We simplify to obtain 5x250x+3y2+229 We remove a common factor of 5 from the first two terms 5(x210x)+3y2+229 and then complete the square to obtain 5(x210x+5252)+3y2+229 This gives 5(x5)2125+3y2+229 or 5(x5)2+3y2+104 Since (x5)20 for all real numbers x and 3y20 for all real numbers y, then the minimum value of 5(x5)2+3y2+104 (and hence of the original expression) is 5(0)+3(0)+104 or 104.
    We note that this value is actually achieved when x=5 (which gives (x5)2=0) and y=0 (which gives 3y2=0).

    Solution 2
    We expand the given expression to obtain (x2+6x+9)+2(y24y+4)+4(x214x+49)+(y2+8y+16) We expand further to obtain x2+6x+9+2y28y+8+4x256x+196+y2+8y+16 The terms involving x are x2+6x+9+4x256x+196=5x250x+205=5x250x+125+80=5(x5)2+80 The terms involving y are 2y28y+8+y2+8y+16=3y2+24 Since (x5)20 for all real numbers x, then the minimum value of 5(x5)2+80 is 80.
    Since y20 for all real numbers y, then the minimum value of 3y2+24 is 24.
    Since the minimum value of (x3)2+4(x7)2 is 80 and the minimum value of 2(y2)2+(y+4)2 is 24, then the minimum value of (x3)2+2(y2)2+4(x7)2+(y+4)2 is 80+24=104.

    Answer: (E)

  23. We label the centres of the coins A,B,D,E,F,G,H, as shown, and we join AB, AD, AE, AF, AG, AH, BD, DE, EF, FG, GH, and HB.

    A circle, labelled A, with six other circles surrounding it, labelled B, D, E, F, G and H when moving around A clockwise. The circles A, B, and G are all medium sized. The circles D, E, and F are all large. The circle H is the smallest. A line is drawn between the centres of every pair of circles that are touching.

    The radius of the circles with centres D, E and F is 3 cm and the radius of the circles with centres A, B and G is 2 cm. (From this point until the very last step of the problem, we will not include the units, which are always centimetres.)
    Let the radius of the circle with centre H be r. We want to determine r.
    When we join the centres of two circles that are just touching, the resulting line segment passes through the point at which the circles touch and the length of this line segment is the sum of the radii of the two circles.
    Thus, AB=2+2=4. Similarly, AG=4, BD=AD=AE=AF=GF=5, DE=EF=6, and HA=HB=HG=r+2.
    Now ADE and AFE each have side lengths 5, 5 and 6, which means that these triangles are congruent.
    Similarly, ADB and AFG are congruent, and ABH and AGH are congruent.
    Since corresponding angles in congruent triangles are equal, then HAB+BAD+DAE is equal to HAG+GAF+FAE.
    But these six angles surround point A, so the sum of these six angles is 360.
    Thus, HAB+BAD+DAE=HAG+GAF+FAE=180.
    Now, we remove the circles from the diagram and focus on the top half of the diagram.
    We join A to K, the midpoint of DE, and D and H to L, the midpoint of AB.

    The top half of the diagram refers to points A, H, B, C, D and E and all their connections.

    Consider ADE.
    Since DE=6 and K is the midpoint of DE, then DK=KE=3.
    Also, since AD=AE=5, then ADE is isosceles and so AK is perpendicular to DE and AK bisects DAE.
    Similarly, AHB is isosceles with HA=HB and ABD is isosceles with DA=DB.
    Since L is the midpoint of AB and AB=4, then AL=LB=2 and DL and HL are perpendicular to AB at L.
    We know that HAB+BAD+DAE=HAL+LAD+2EAK=180.
    But sin(EAK)=EKAE=35, so EAK36.87.
    Also, cos(LAD)=ALAD=25, so LAD66.42.
    Thus, HAL=180LAD2EAK18066.422(36.87)39.84.
    Finally, we know that cos(HAL)=ALHA=2r+2.
    Since cos(39.84)0.7679, then 2r+20.7679.
    From this we obtain r20.76792.Since20.767920.60451, then of the given choices, this radius is closest to 0.605 cm.
    (If more decimal places are carried through from earlier calculations, we obtain r0.60466, which is still closest to 0.605 cm.)

    Answer: (D)

  24. We begin by understanding and describing k in a different way.
    Each positive integer k can be placed between two consecutive perfect squares. More precisely, for each positive integer k, there exists a unique positive integer n with n2k<(n+1)2.
    Since n2k<(n+1)2, then n2k<n2+2n+1 and nk<n+1.
    Since n and n+1 are consecutive integers, then k=n.
    In other words, n is the largest positive integer less than or equal to k.
    So we want to calculate the sum of all integers k with 1k999999 for which k is a multiple of its corresponding n.
    Now n2=nn and n2+2n+1=n(n+2)+1.
    This means that the possible values of k with n2k<n2+2n+1 are the multiples of n in this range, or k=nn, k=n(n+1), and k=n(n+2).
    Since k999999, then k<10002=1000000, and so n999.
    So the given problem is equivalent to determining the sum of n2, n(n+1) and n(n+2) for each n from 1 to 999.
    Since n2+n(n+1)+n(n+2)=3n2+3n, then we want to determine the sum of 3n2+3n for n=1 to n=999 inclusive.
    Thus, S=(3(12)+3(1))+(3(22)+3(2))++(3(9982)+3(998))+(3(9992)+3(999))=3(12+22++9982+9992)+3(1+2++998+999) Since 1+2++(m1)+m=m(m+1)2 and 12+22++(m1)2+m2=m(m+1)(2m+1)6 for every positive integer m, then S=3999(1000)(1999)6+3999(1000)2=3(999)(1000)6(1999+3)=999(1000)2(2002)=999(1000)(1001) and so S=999999000.

    Answer: (C)

  25. We begin by partitioning the set A into disjoint subsets of the form Pb={b,3b,9b,,3kb} where b is a positive integer with 1b2045 and k is the largest non-negative integer with 3kb2045.
    The first two of these sets are P1={1,3,9,27,81,243,729} and P2={2,6,18,54,162,486,1458}.
    These sets have b=1 and b=2, and thus k=6.
    We will see that each element of A is an element of exactly one of these sets.
    Since 37=2187, then 37b2187>2045 for every positive integer b, so k=6 is the largest possible value of k that we can have.
    For each k with 0k6, we determine the range of values of b that give that k:

    Since we want to create disjoint sets Pb whose union is the set A={1,2,3,,2044,2045}, then we exclude all b’s that are multiples of 3. (If b were a multiple of 3, it would appear as an element of a set Pr with rb.)
    For each of the ranges above, we count the number of multiples of 3 that we need to exclude:

    k Range of values of b # of multiples of 3 # of remaining b # of elements in each Pb
    6 1b2 0 20=2 7
    5 3b8 2 62=4 6
    4 9b25 6 176=11 5
    3 26b75 17 5017=33 4
    2 76b227 50 15250=102 3
    1 228b681 152 454152=302 2
    0 682b2045 454 1364454=910 1

    Note for example that the range 26b75 contains 7525=50 integers and includes the multiples of 3 from 93 to 253, inclusive, so includes 258=17 multiples of 3, and the range 228b681 contains 681227=454 integers and includes the multiples of 3 from 763 to 2273, inclusive, so includes 22775=152 multiples of 3.
    An integer b in A that is not a multiple of 3 generates a set Pb and cannot appear in a different set Pr. Any multiple of 3, say m, will appear in exactly one Pb where the value of b (which is itself not a multiple of 3) is obtained by dividing out all of factors of 3 from m.
    We note that 27+46+115+334+1023+3022+9101=2045 so the union of these sets Pb includes enough elements to be the entire set A. No element of A appears in more than one Pb, so the union of these sets Pb is the entire set A.
    Why are these sets useful in the given problem?
    A subset of A is triple-free if it does not contain two elements, one of which is three times the other.
    This means that a subset of A is triple-free exactly when it does not contain two consecutive elements of any Pb.
    A triple-free subset T of A that contains as many elements as possible will contain as many elements as possible from each of the Pb defined above.
    Since no two consecutive elements of any Pb can appear in T, then a Pb with 7 elements can contribute at most 4 elements to T (every other element starting with the first), a Pb with 6 elements can contribute at most 3 elements to T (it cannot contain 4 elements without having two consecutive elements), and so on.
    For each k from k=7 to k=0, each Pb above can contribute at most the following number of elements to T:

    k # of elements in each Pb Number of elements that can be chosen
    6 7 4
    5 6 3
    4 5 3
    3 4 2
    2 3 2
    1 2 1
    0 1 1

    This means that T can contain at most 24+43+113+332+1022+3021+9101=1535 elements. This agrees with the information in the problem statement.
    Where is there choice in creating T?
    If a Pb contains 1 element, then this element must be chosen for T.
    If a Pb contains 3 elements, then there is 1 way to choose 2 elements for T, since in this case the 1st and 3rd elements (in increasing order) must be chosen.
    If a Pb contains 5 elements, then there is 1 way to choose 3 elements for T, since in this case the 1st, 3rd and 5th elements (in increasing order) must be chosen.
    If a Pb contains 7 elements, then there is 1 way to choose 4 elements for T, since in this case the 1st, 3rd, 5th and 7th (in increasing order) elements must be chosen.
    If a Pb contains 2 elements, then there are 2 ways to choose 1 element for T.
    If a Pb contains 4 elements, then there are 3 ways to choose 2 elements for T. (If we want to choose 2 elements from {A,B,C,D} without choosing consecutive elements, we can choose A and C, or A and D, or B and D.)
    If a Pb contains 6 elements, then there are 4 ways to choose 3 elements for T. (If we want to choose 3 elements from {A,B,C,D,E,F} without choosing consecutive elements, we can choose A and C and E, or A and C and F, or A and D and F, or B and D and F.)
    This means that the number of ways of choosing a triple-free subset T of A with as many elements as possible is 1244111333110223021910.
    This is because there are 2 sets Pb with k=6 and 1 way of choosing 4 elements from each of them (giving 12 choices in total), 4 set Pb with k=5 and 4 ways of choosing 3 elements from each of them (giving 44 choices in total), and so on.
    The seven factors in this product give, from k=6 to k=0, the number of ways of choosing the maximum number of elements from a set Pb corresponding to that value of k raised to the power of the number of such sets Pb.
    Therefore, the number of ways equals 443332302=2310333.
    From the statement of the problem, we thus have N=22+32+3102+332=97202.
    The final three digits of this integer are 202.

    Answer: (A)